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Claim. Theorem 1.2 of T. Sanders, The Erdős--Moser sum-free set problem, Canad. J. Math. 73 (2021), no. 1, 63--107, p. 2: "For every finite set of integers $A$ we have $M(A)=\log^{1+\Omega(1)}|A|$", where $M(A)$ is the largest size of a set $S\subset A$ whose restricted sumset $\{s+s':s,s'\in S,\ s\ne s'\}$ is disjoint from $A$. The abstract states it as an absolute constant $c>0$ with $M(A)\ge\log_3^{1+c}|A|$ for every finite $A$, footnote 2 reconciling the two forms for small $|A|$. In the notation of Problem 787,

$$g(n)\ge(\log n)^{1+c}$$

for an absolute $c>0$ and all large $n$. The theorem is stated for finite sets of integers; Choi's observation that the real-set function equals the integer one, which the problem page records, carries it to the site's sets of real numbers. The proof strengthens the Sudakov--Szemerédi--Vu strategy with Proposition 2.7, which replaces their fivefold exponential dependence by $\exp(k^{C+o(1)})$. Cited as [Sa21] on the problem page. Library home sanders_2021_erdos_moser_sum_free_set_problem; result page Theorem 1.2.

Covers. The lower bound $g(n)\ge(\log n)^{1+c}$ for an absolute $c>0$, the first lower bound a power of $\log n$ beyond the first. Not covered: the order of growth of $g(n)$, which lies between this bound and Ruzsa's $\exp(O(\sqrt{\log n}))$.

Depends on. Choi's 1971 paper, whose reduction of the real-set problem to sets of integers carries the theorem to the site's formulation.

Acceptance. Refereed: the paper is the publisher's version of record in the Canadian Journal of Mathematics (the Crossref record gives online publication on 23 September 2019, in the February 2021 issue); the page is named by the first arXiv version, 1804.03356v1 of 10 April 2018. The site's curator, Thomas F. Bloom, credits the lower bound to Sanders in the problem page's commentary (label OPEN, page last edited 23 January 2026); the problem is not marked settled there, so the credit is recorded here and is not listed as reviewed. The statement is quoted from arXiv v3 of 31 July 2019; the proof is not checked in this corpus.